I was looking at feats for my backup character, and was wondering:
Would the Spell Sniper feat affect the range of a Dragonborn's racial breath weapon?
No, it wouldn't
The Spell Sniper feat (PHB p. 170) includes this ability:
When you cast a spell that requires you to make an attack roll, the spell's range is doubled.
The dragonborn's Breath Weapon trait (PHB p. 34) is not a spell. If it were a spell, it would either refer to an existing spell (in italics, typically) or explicitly say it is a spell. (See: What counts as a spell?) It is not even magical, because it does not refer to any spells and it does not say it is magical.
The Breath Weapon ability also does not involve an attack roll. You just pick an area originating from yourself, and enemies in that area make a saving throw. There is no mention of an attack roll.
As such, on account of not being a spell and not involving an attack roll, the Breath Weapon's range is not affected by Spell Sniper. Both conditions would need to be met for Spell Sniper to apply.